Properties

Label 2-1400-1.1-c1-0-9
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.363·3-s + 7-s − 2.86·9-s + 5.14·11-s − 4.64·13-s + 3.86·17-s − 0.778·19-s + 0.363·21-s + 5.00·23-s − 2.13·27-s + 9.42·29-s + 4.72·31-s + 1.86·33-s + 6·37-s − 1.68·39-s − 1.00·41-s − 7.00·43-s − 11.4·47-s + 49-s + 1.40·51-s + 7.55·53-s − 0.282·57-s + 12.5·59-s + 11.5·61-s − 2.86·63-s − 11.7·67-s + 1.82·69-s + ⋯
L(s)  = 1  + 0.209·3-s + 0.377·7-s − 0.955·9-s + 1.55·11-s − 1.28·13-s + 0.938·17-s − 0.178·19-s + 0.0792·21-s + 1.04·23-s − 0.410·27-s + 1.74·29-s + 0.848·31-s + 0.325·33-s + 0.986·37-s − 0.270·39-s − 0.157·41-s − 1.06·43-s − 1.66·47-s + 0.142·49-s + 0.196·51-s + 1.03·53-s − 0.0374·57-s + 1.62·59-s + 1.47·61-s − 0.361·63-s − 1.43·67-s + 0.219·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.890180044\)
\(L(\frac12)\) \(\approx\) \(1.890180044\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 - 0.363T + 3T^{2} \)
11 \( 1 - 5.14T + 11T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 - 3.86T + 17T^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
23 \( 1 - 5.00T + 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 - 4.72T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 1.00T + 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 7.55T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 - 5.00T + 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 4.67T + 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + 1.58T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.606032334057656756392825741345, −8.648048244258167733590530724345, −8.153153746814121881136114579261, −7.05424072649334404947060653047, −6.39766076663441368803609041224, −5.30534279764248165983535326127, −4.54379647525295333219030650748, −3.38255358477665102765446509456, −2.47191537705500274173708198819, −1.03381490661590609565760309995, 1.03381490661590609565760309995, 2.47191537705500274173708198819, 3.38255358477665102765446509456, 4.54379647525295333219030650748, 5.30534279764248165983535326127, 6.39766076663441368803609041224, 7.05424072649334404947060653047, 8.153153746814121881136114579261, 8.648048244258167733590530724345, 9.606032334057656756392825741345

Graph of the $Z$-function along the critical line